Slightly supercritical percolation on nonamenable graphs II: growth and isoperimetry of infinite clusters

Abstract

We study the growth and isoperimetry of infinite clusters in slightly supercritical Bernoulli bond percolation on transitive nonamenable graphs under the $L^2$ boundedness condition ($p_c<p_{2\rightarrow 2}$). Surprisingly, we find that the volume growth of infinite clusters is always purely exponential (that is, the subexponential corrections to growth are bounded) in the regime $p_c<p<p_{2\rightarrow 2}$, even when the ambient graph has unbounded corrections to exponential growth. For p slightly larger than $p_c$, we establish the precise estimates

$$\begin{aligned} \textbf{E}_p \left[ \# B_\textrm{int}(v,r) \right]&\asymp \left( r \wedge \frac{1}{p-p_c} \right) e^{\gamma _\textrm{int}(p) r} \\ \textbf{E}_p \left[ \# B_\textrm{int}(v,r) \mid v \leftrightarrow \infty \right]&\asymp \left( r \wedge \frac{1}{p-p_c} \right) ^2 e^{\gamma _\textrm{int}(p) r} \end{aligned}$$

for every $v\in V$, $r \ge 0$, and $p_c < p \le p_c+\delta$, where the growth rate $\gamma _\textrm{int}(p) = \lim \frac{1}{r} \log \textbf{E}_p\#B(v,r)$ satisfies $\gamma _\textrm{int}(p) \asymp p-p_c$. We also prove a percolation analogue of the Kesten–Stigum theorem that holds in the entire supercritical regime and states that the quenched and annealed exponential growth rates of an infinite cluster always coincide. We apply these results together with those of the first paper in this series to prove that the anchored Cheeger constant of every infinite cluster K satisfies

$$\begin{aligned} \frac{(p-p_c)^2}{\log [1/(p-p_c)]} \preceq \Phi ^*(K) \preceq (p-p_c)^2 \end{aligned}$$

almost surely for every $p_c<p\le 1$.